470 MATHEMATICS 



by laying down explicitly a list of logical conceptions and prin- 

 ciples which alone are to be used; and, secondly, he insists, 1 on the 

 contrary, that no mathematical system, to use again the technical 

 term introduced above, be studied in pure mathematics whose exist- 

 ence cannot be established solely from the logical principles on which 

 all mathematics is based. Inasmuch as the development of mathemat- 

 ics during the last fifty years has shown that the existence of most, 

 if not all the mathematical systems which have proved to be im- 

 portant can be deduced when once the existence of positive integers 

 is granted, the point about which interest must centre here is the 

 proof, which Russell attempts, of the existence of this latter sys- 

 tem. 2 This proof will necessarily require that, among the logical 

 principles assumed, existence theorems be found. Such theorems 

 do not seem to be explicitly stated by Russell, the existence theorems 

 which make their appearance further on being evolved out of some- 

 what vague philosophical reasoning. There are also other reasons, 

 into which I cannot enter here, why I am not able to regard the 

 attempt made in this direction by Russell as completely successful. 3 

 Nevertheless, in view of the fact that the system of finite positive 

 integers is necessary in almost all branches of mathematics (we 

 cannot speak of a triangle or a hexagon without having the numbers 

 three and six at our disposal), it seems extremely desirable that the 

 system of logical principles which we lay at the foundation of all 

 mathematics be assumed, if possible, broad enough so that the 

 existence of positive integers - - at least finite integers - - follows from 

 it; and there seems little doubt that this can be done in a satisfactory 

 manner. When this has been done we shall perhaps be able to regard, 

 with Russell, pure mathematics as consisting exclusively of deduc- 

 tions "by logical principles from logical principles." 



VIII. The Non-Deductive Elements in Mathematics 



I fear that many of you will think that what I have been saying 

 is of an extremely one-sided character, for I have insisted merely on 

 the rigidly deductive form of reasoning used and the purely abstract 

 character of the objects considered in mathematics. These, to the 

 great majority of mathematicians, are only the dry bones of the 

 science. Or, to change the simile, it may perhaps be said that instead 

 of inviting you to a feast I have merely shown you the empty dishes 



1 In the formal definition of mathematics at the beginning of the book this is 

 not stated or in any way implied; and yet it comes out so clearly throughout 

 the book that this is a point of view which the author regards as essential, that 

 I have not hesitated to include it as a part of his definition. 



2 Cf. also Burali-Forti, Congres Internationale de philosophic. Paris, A'ol. in, 

 p. 280. 



J Russell's unequivocal repudiation of nominalism in mathematics seems to 

 me a serious if not an insurmountable barrier to progress. 



